On the space of projective curves of maximal regularity

Abstract

Let \(\Gamma _{r,d}\) be the space of smooth rational curves of degree d in \({\mathbb {P}}^r\) of maximal regularity. Then the automorphism group \(\mathrm{Aut}({\mathbb {P}}^r)=\mathrm{PGL}(r+1)\) acts naturally on \(\Gamma _{r,d}\) and thus the quotient \(\Gamma _{r,d}/ \mathrm{PGL}(r+1)\) classifies those rational curves up to projective motions. In this paper, we show that \(\Gamma _{r,d}\) is an irreducible variety of dimension \(3d+r^2-r-1\). The main idea of the proof is to use the canonical form of rational curves of maximal regularity which is given by the \((d-r+2)\)-secant line. Also, through the geometric invariant theory, we discuss how to give a scheme structure on the \(\mathrm{PGL}(r+1)\)-orbits of rational curves.